91Ó°ÊÓ

Let \(f(t): I \rightarrow V_{2}\) be given by \(f_{1}(t)=t^{2}-t, f_{2}(t)=t^{3}-3 t\), with \(I=\\{t:-3 \leqslant\) \(t \leqslant 3\\}\). Decide whether or not the path \(\Gamma\) in \(\mathbb{R}^{2}\) represented by \(f\) is an arc.

In each of Problems 9 through 12 find \(\boldsymbol{T}, \boldsymbol{N}, \boldsymbol{B}, \kappa\), and \(\tau\) at the given value of \(t\) for the given function \(\boldsymbol{f}: \mathbb{R}^{1} \rightarrow V_{3}\). $$ f(t)=(1 / 3) t^{3} e_{1}+2 t e_{2}+(2 / t) e_{3}, \quad t=2 $$

(a) Show that any half-plane in \(\mathbb{R}^{2}\) is convex. (b) Show that in \(\mathbb{R}^{N}\) the half-space given by \(x_{N} \geqslant 0\) is convex.

Find a formula for curl(curl \(u\) ) in terms of \(\left(x_{1}, x_{2}, x_{3}\right)\) and \(\left(e_{1}, e_{2}, e_{3}\right)\) if \(u=u_{1} e_{1}+\) \(u_{2} e_{2}+u_{3} e_{3}\) and each of the \(u_{i}\) is a \(C^{2}\) function on \(\mathbb{R}^{3}\).